The Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) will commence full-scale operations in 2026, yielding an unprecedented volume of astronomical images. Constructing an astronomical catalog, a table of imaged stars, galaxies, and their properties, is a fundamental step in most scientific workflows based on astronomical image data. Traditional deterministic cataloging methods lack statistical coherence as cataloging is an ill-posed problem, while existing probabilistic approaches suffer from computational inefficiency, inaccuracy, or the inability to perform inference with multiband coadded images, the primary output format for LSST images. In this article, we explore a recently developed Bayesian inference method called neural posterior estimation (NPE) as an approach to cataloging. NPE leverages deep learning to achieve both computational efficiency and high accuracy. When evaluated on the DC2 Simulated Sky Survey -- a highly realistic synthetic dataset designed to mimic LSST data -- NPE systematically outperforms the standard LSST pipeline in light source detection, flux measurement, star/galaxy classification, and galaxy shape measurement. Additionally, NPE provides well-calibrated posterior approximations. These promising results, obtained using simulated data, illustrate the potential of NPE in the absence of model misspecification. Although some degree of model misspecification is inevitable in the application of NPE to real LSST images, there are a variety of strategies to mitigate its effects.